# On De Giorgi’s conjecture in dimension \$N\ge 9\$ | Annals of Mathematics

Abstract A celebrated conjecture due to De Giorgi states that any bounded solution of the equation Δu+(1−u2)u=0inℝN with ∂yNu>0 must be such that its level sets {u=λ} are all hyperplanes, at least for dimension N≤8. A counterexample for N≥9 has long been believed to exist. Starting from a minimal graph Γ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in ℝN, N≥9, we prove that for any small α>0 there is a bounded solution uα(y) with ∂yNuα>0, which resembles tanh(t2√), where t=t(y) denotes a choice of signed distance to the blown-up minimal graph Γα:=α−1Γ. This solution is a counterexample to De Giorgi’s conjecture for N≥9.

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### 数学年刊（Annals of Mathematics）

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